On the long time behavior of the stochastic heat equation

• 成果类型: Article
• 署名作者: Bertini, L; Giacomin, G
• 署名单位: Sapienza University Rome; University of Zurich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050226
• 发表日期: 1999
• 页码: 279-289
• 关键词: particle-systems; interfaces

摘要

We consider the stochastic heat equation in one space dimension and compute - for a particular choice of the initial datum - the exact long time asymptotic. In the Carmona-Molchanov approach to intermittence in non stationary random media this corresponds to the identification of the sample Lyapunov exponent, Equivalently, by interpreting the solution as the partition function of a directed polymer in a random environment, we obtain a weak law of large numbers for the quenched free energy. The result agrees with the one obtained in the physical literature via the replica method. The proof is based on a representation of the solution in terms of the weakly asymmetric exclusion process.